With it, we can form the GNS representation \(\pi _{\omega }\)  [1]}
of the algebra (that is, \(\left(a,b\right)_{\mathcal {H}_{\omega }}\doteq \omega (a^{*}b)=\left(a^{*}b\right)_{0}=\sum _{r\in \mathbb {Z}^{2}}a_{r}^{*}b_{r}\) .)
We denote as \(\underline{a}\)  to the algebra elements when seen as
elements in the Hilbert space \(\mathcal {H}_{\omega }\)  of the representation
(whose cyclic vector is \(\underline{I}\)  and is irreducible) and the
same for the derivations when seen as acting on the algebra representation.
We can now define a Dirac operator by setting (on the dense domain
\(\underline{\mathcal {\widehat{A}}}_{\theta (P_{N_{1}},P_{N_{2}})}\) )
\(D_{P_{N_{1}}P_{N_{2}}}\doteq -i\mathrm {a}(P_{N_{1}},P_{N_{2}})^{-\frac{1}{2}}\left[\sigma _{1}\underline{\delta }_{1}+\sigma _{2}\underline{\delta }_{2}\right],\) 
